More About Angles

Before proceeding further into study of angles, we need to learn some definitions & conventions used in describing angles. We need them to communicate geometrical ideas and study theorems & constructions in Plane Geometry.

Definitions

Ray - Ray is defined as a line starting from a given point, going in any direction, indefinitely. Geometers usually define an angle with reference to 2 rays starting from the same point and going in different directions. Euclid however defines an angle without using the idea of rays.

A Ray can be thought of as half a line. The line has no beginning & no end. A Ray has a beginning but no end. Rays of Light emerging from a torch can be thought of as rays.

Arms– The two lines (or rays) which form the angle are called the arms of the angle. The arms are imagined to extend forever. For convenience of labelling, the arms are assumed to be of a certain arbitrary magnitude.

Sides – In a closed figure like a triangle, the two arms which enclose any of the angles in the triangle are also called Sides. In closed figures, the side also refers to the length of the line segment. Hence, we can say the sum of any 2 sides of a triangle is more than the 3rdside.

If the end points of the side of a closed figure are points A & B, then the side is referred to as AB.

Vertex– The point at which the arms of the angle meet is called the Vertex of the angle. It is also the point about which the arms rotate when forming any angle. “Vertices” is the plural form.

What does the angle consist of?

Geometrically an angle consists only of the two arms (sides) of the angle and the vertex where they meet.

Any angle, divides the plane on which it is drawn into 2 areas, the interior & exterior

Interior of an angle– The area which lies within the arms of an angle is called the Interior of the angle. Since the arms can be extended infinitely, the interior of an angle also extends infinitely. Though we say interior “of an angle” it does not form part of an angle.

Exterior of an angle– The area outside the arms of an angle is called the Exterior of the angle.

Name of an Angle- If the two arms of an angle are named OA & OB, with O being the vertex, the angle is called Angle AOB or Angle BOA. By convention upper case letters are used to name an angle.

Measure of an Angle

How do we measure the angle or “amount of rotation”?

One of the most natural rotational movements was a “complete rotation” by which a rotating object comes back to the starting position; much like the daily movement of the Sun and Moon across the sky every day. Another familiar experience for children is the movement of the Second Hand of a clock, which comes back to 12 every hour.

The complete rotation, which was called Complete Angle, was taken as the standard to measure angles. All angles could be expressed as a fraction of the Complete Angle.

What was the measure of the Complete Angle?

The Complete Angle was made equal to the number of days the Earth takes to make one revolution around the Sun. In early days it was thought of as 360 days and hence the complete angle was considered as made of 360 degrees.

Degree was the unit for measurement of angles. We should not confuse it with another unit by the same name which is used to measure temperature!

Angle Measure – Radian

Angles are also measured in units called radians. The radian is intimately connected with the other properties of a circle. The radian is defined as the angle subtended at the centre of a circle by an arc of a circle, which is equal to the radius of the circle in length.

In radians the complete angle is 2π & the sum of the angles of a triangle is π. Use of radians instead of degrees make most formulas in trigonometry and advanced math, simple and elegant.